1. Field of Invention
The present application is directed toward systems and methods for employing a finite difference scheme for simulating the evaporation of a droplet on a domain whose size is varying over time.
2. Description of Related Art
The industrial printing process includes the production of small ink droplets. Each ink droplet may contain one or more solvents and solutes. Such solute is a metal, polymer, other suitable material, or mixtures of such materials. Each ink droplet may be ejected onto a target area of a patterned substrate. After the droplets land, the solvent(s) evaporate(s) and a thin film of the solute is formed. Controlling the final pattern of the solute film is important to maintaining the quality and repeatability of the printing process. In order to control the final pattern of the solute film, it is important to understand how the final pattern is formed. To do so, it is important to understand the influence of factors that influence and control the formation of the final pattern, such as evaporation rate, initial droplet volume and shape, initial solute concentration, and contact line dynamics. Numerical simulations of the printing process are useful tools for understanding the influence of these factors and for developing the control process for printing.
In the later stage of the ink drying process, the aspect ratio of the droplet (the length of the droplet vs. its height) increases and becomes quite large. Lubrication theory, which is good for describing the physics of thin films, may be applied to describe the evaporation physics and greatly reduce the complexity of the simulation at this later stage. Lubrication theory is an approximation of the Navier-Stokes equation for thin films. The application of lubrication theory to droplet evaporation results in two equations: a fourth-order interface evolution equation that describes the evolution of droplet surface (or interface) considering the effects of evaporation rate, surface tension, and fluid viscosity; and second-order solute convection/diffusion equation that describes the motion of solute particles under the influence of fluid velocity and particle diffusion.
As observed in experiments, the evaporation of solvent drives solute to move toward the contact line regions where it accumulates. These regions and a center region of the droplet soon reach saturated concentration. Hence, the evaporation ceases in those regions. Therefore, the fluid equations remain valid only for the regions where the solute has not reached saturated concentration.
The present invention improves on previous work on finite difference methods for solving lubrication equations for droplet evaporation. Previously, the location of the contact line was fixed, as was the solution domain. Even if the regions near the contact line reach saturation, there is still a non-zero flow field in that region. The flow, though small, could introduce an error in simulation results in film thickness, especially when the solute concentration is high near the end of the evaporation process.